Total variation bound for Hadwiger's functional using Stein's method

Let $K$ be a convex body in ${\mathbb{R}}^d$. Let $X_K$ be a $d$-dimensional random vector distributed according to the Hadwiger-Wills density ${\mu }_K$ associated with $K$, defined as ${\mu }_K(x)=c{e}^{-\pi {\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}^2(x,K)}$, $x\in {\mathbb{R}}^d$. Finally, let the information content $H_K$ be defined as $H_K={\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}^2(X_K,K)$. The goal of this paper is to study the fluctuations of $H_K$ around its expectation as the dimension $d$ go to infinity. Relying on Stein’s method and Brascamp-Lieb inequality, we compute an explicit bound for the total variation distance between $H_K$ and its Gaussian counterpart.